Agglomeration-Based Geometric Multigrid Solvers for Compact Discontinuous Galerkin Discretizations on Unstructured Meshes

We present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.

[1]  S. Dargaville,et al.  A comparison of element agglomeration algorithms for unstructured geometric multigrid , 2020, ArXiv.

[2]  Hari Sundar,et al.  Comparison of multigrid algorithms for high‐order continuous finite element discretizations , 2014, Numer. Linear Algebra Appl..

[3]  ROBERT I. SAYE,et al.  Efficient multigrid solution of elliptic interface problems using viscosity-upwinded local discontinuous Galerkin methods , 2019, Communications in Applied Mathematics and Computational Science.

[4]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[5]  Will Pazner,et al.  Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods , 2019, SIAM J. Sci. Comput..

[6]  Martin Berggren,et al.  Agglomeration Multigrid for the Vertex-Centered Dual Discontinuous Galerkin Method , 2010 .

[7]  João Luiz F. Azevedo,et al.  On the development of an agglomeration multigrid solver for turbulent flows , 2003 .

[8]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[9]  David L. Darmofal,et al.  p-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier-Stokes equations , 2005 .

[10]  V. E. Henson,et al.  BoomerAMG: a parallel algebraic multigrid solver and preconditioner , 2002 .

[11]  Jinchao Xu,et al.  Iterative Methods by Space Decomposition and Subspace Correction , 1992, SIAM Rev..

[12]  Jinchao Xu,et al.  An agglomeration multigrid method for unstructured grids , 1998 .

[13]  Stefano Zampini,et al.  MFEM: a modular finite element methods library , 2019, 1911.09220.

[14]  Per-Olof Persson,et al.  Newton-GMRES Preconditioning for Discontinuous Galerkin Discretizations of the Navier--Stokes Equations , 2008, SIAM J. Sci. Comput..

[15]  Jim E. Jones,et al.  AMGE Based on Element Agglomeration , 2001, SIAM J. Sci. Comput..

[16]  Per-Olof Persson,et al.  The Compact Discontinuous Galerkin (CDG) Method for Elliptic Problems , 2007, SIAM J. Sci. Comput..

[17]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[18]  George Biros,et al.  A Parallel Geometric Multigrid Method for Finite Elements on Octree Meshes , 2010, SIAM J. Sci. Comput..

[19]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[20]  Vipin Kumar,et al.  A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs , 1998, SIAM J. Sci. Comput..

[21]  Dimitri J. Mavriplis,et al.  A Parallel hp-Multigrid Solver for Three-Dimensional Discontinuous Galerkin Discretizations of the Euler Equations , 2007 .

[22]  Ludmil T. Zikatanov,et al.  Algebraic multigrid methods * , 2016, Acta Numerica.